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Symmetric equilibria for a beam with a nonlinear elastic foundation. (English) Zbl 0815.34014
Using a standard variational technique due to P. Rabinowitz, the authors study the following boundary value problem which connects with the elastic beam theory: \(u^{(4)} (x) + g(x,u(x)) = 0\), \(u''(0) = u''(1) = 0\), \(u'''(0) = - f(u(0))\), \(u'''(1) = f(u(1))\). Under some natural restrictions on \(f\) and \(g\), having mechanical nature, they establish existence of solutions to the problem in \(H^ 2_ s(0,1)\), a subspace of symmetrical functions \(u \in H^ 2_ s(0,1)\), \(u(x) = u(1-x)\).

34B15 Nonlinear boundary value problems for ordinary differential equations
74K10 Rods (beams, columns, shafts, arches, rings, etc.)
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